1. Bits and Bytes January 17, 2002 Topics Why bits? Representing information as bits –Binary/Hexadecimal –Byte representations »numbers »characters and strings »Instructions Bit-level manipulations –Boolean algebra –Expressing in C Suggested Reading: 2.1 Suggested Problems: 2.35 and 2.37 class02.ppt 15-213 CS 213 S’02 (Based on CS 213 F’01)

2. – 2 – class02.ppt Why Don’t Computers Use Base 10? Base 10 Number Representation That’s why fingers are known as “digits” Natural representation for financial transactions –Floating point number cannot exactly represent $1.20 Even carries through in scientific notation –1.5213 X 10 4 Implementing Electronically Hard to store –ENIAC (First electronic computer) used 10 vacuum tubes / digit Hard to transmit –Need high precision to encode 10 signal levels on single wire Messy to implement digital logic functions –Addition, multiplication, etc.

3. CS 213 S’02 (Based on CS 213 F’01) – 3 – class02.ppt Binary Representations Base 2 Number Representation Represent 15213 10 as 11101101101101 2 Represent 1.20 10 as 1.0011001100110011[0011]… 2 Represent 1.5213 X 10 4 as 1.1101101101101 2 X 2 13 Electronic Implementation Easy to store with bistable elements Reliably transmitted on noisy and inaccurate wires Straightforward implementation of arithmetic functions 0.0V 0.5V 2.8V 3.3V 010

4. CS 213 S’02 (Based on CS 213 F’01) – 4 – class02.ppt Byte-Oriented Memory Organization Programs Refer to Virtual Addresses Conceptually very large array of bytes Actually implemented with hierarchy of different memory types –SRAM, DRAM, disk –Only allocate for regions actually used by program In Unix and Windows NT, address space private to particular “process” –Program being executed –Program can clobber its own data, but not that of others Compiler + Run-Time System Control Allocation Where different program objects should be stored Multiple mechanisms: static, stack, and heap In any case, all allocation within single virtual address space

5. CS 213 S’02 (Based on CS 213 F’01) – 5 – class02.ppt Encoding Byte Values Byte = 8 bits Binary 00000000 2 to 11111111 2 Decimal: 0 10 to 255 10 Hexadecimal 00 16 to FF 16 –Base 16 number representation –Use characters ‘0’ to ‘9’ and ‘A’ to ‘F’ –Write FA1D37B 16 in C as 0xFA1D37B »Or 0xfa1d37b 000000 110001 220010 330011 440100 550101 660110 770111 881000 991001 A101010 B111011 C121100 D131101 E141110 F151111 Hex Decimal Binary

6. CS 213 S’02 (Based on CS 213 F’01) – 6 – class02.ppt Machine Words Machine Has “Word Size” Nominal size of integer-valued data –Including addresses Most current machines are 32 bits (4 bytes) –Limits addresses to 4GB –Becoming too small for memory-intensive applications High-end systems are 64 bits (8 bytes) –Potentially address  1.8 X 10 19 bytes Machines support multiple data formats –Fractions or multiples of word size –Always integral number of bytes

7. CS 213 S’02 (Based on CS 213 F’01) – 7 – class02.ppt Word-Oriented Memory Organization Addresses Specify Byte Locations Address of first byte in word Addresses of successive words differ by 4 (32-bit) or 8 (64-bit) 0000 0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 32-bit Words BytesAddr. 0012 0013 0014 0015 64-bit Words Addr = 0000 Addr = 0008 Addr = 0000 Addr = 0004 Addr = 0008 Addr = 0012

8. CS 213 S’02 (Based on CS 213 F’01) – 8 – class02.ppt Data Representations Sizes of C Objects (in Bytes) C Data TypeCompaq AlphaTypical 32-bitIntel IA32 int 444 long int 844 char 111 short 222 float 444 double 888 long double 8810/12 char * 844 »Or any other pointer

9. CS 213 S’02 (Based on CS 213 F’01) – 9 – class02.ppt Byte Ordering Issue How should bytes within multi-byte word be ordered in memory Conventions Alphas, PC’s are “Little Endian” machines –Least significant byte has lowest address Sun’s, Mac’s are “Big Endian” machines –Least significant byte has highest address Example Variable x has 4-byte representation 0x01234567 Address given by &x is 0x100 0x1000x1010x1020x103 01234567 0x1000x1010x1020x103 67452301 Big Endian Little Endian

10. CS 213 S’02 (Based on CS 213 F’01) – 10 – class02.ppt Examining Data Representations Code to Print Byte Representation of Data Casting pointer to unsigned char * creates byte array typedef unsigned char *pointer; void show_bytes(pointer start, int len) { int i; for (i = 0; i < len; i++) printf("0x%p\t0x%.2x\n", start+i, start[i]); printf("\n"); } Printf directives: %p :Print pointer %x :Print Hexadecimal

11. CS 213 S’02 (Based on CS 213 F’01) – 11 – class02.ppt show_bytes Execution Example int a = 15213; printf("int a = 15213;\n"); show_bytes((pointer) &a, sizeof(int)); Result: int a = 15213; 0x11ffffcb80x6d 0x11ffffcb90x3b 0x11ffffcba0x00 0x11ffffcbb0x00

12. CS 213 S’02 (Based on CS 213 F’01) – 12 – class02.ppt Representing Integers int A = 15213; int B = -15213; long int C = 15213; Decimal: 15213 Binary: 0011 1011 0110 1101 Hex: 3 B 6 D 6D 3B 00 Linux/Alpha A 3B 6D 00 Sun A 93 C4 FF Linux/Alpha B C4 93 FF Sun B Two’s complement representation (Covered next lecture) 00 6D 3B 00 Alpha C 3B 6D 00 Sun C 6D 3B 00 Linux C

13. CS 213 S’02 (Based on CS 213 F’01) – 13 – class02.ppt Representing Pointers int B = -15213; int *P = &B; Alpha Address Hex: 1 F F F F F C A 0 Binary: 0001 1111 1111 1111 1111 1111 1100 1010 0000 01 00 A0 FC FF Alpha P FB 2C EF FF Sun P Sun Address Hex: E F F F F B 2 C Binary: 1110 1111 1111 1111 1111 1011 0010 1100 Different compilers & machines assign different locations to objects FF BF D4 F8 Linux P Linux Address Hex: B F F F F 8 D 4 Binary: 1011 1111 1111 1111 1111 1000 1101 0100

14. CS 213 S’02 (Based on CS 213 F’01) – 14 – class02.ppt Representing Floats Float F = 15213.0; IEEE Single Precision Floating Point Representation Hex: 4 6 6 D B 4 0 0 Binary: 0100 0110 0110 1101 1011 0100 0000 0000 15213: 1110 1101 1011 01 Not same as integer representation, but consistent across machines 00 B4 6D 46 Linux/Alpha F B4 00 46 6D Sun F

15. CS 213 S’02 (Based on CS 213 F’01) – 15 – class02.ppt char S[6] = "15213"; Representing Strings Strings in C Represented by array of characters Each character encoded in ASCII format –Standard 7-bit encoding of character set –Other encodings exist, but uncommon –Character “0” has code 0x30 »Digit i has code 0x30 +i String should be null-terminated –Final character = 0 Compatibility Byte ordering not an issue –Data are single byte quantities Text files generally platform independent –Except for different conventions of line termination character! Linux/Alpha S Sun S 32 31 35 33 00 32 31 35 33 00

16. CS 213 S’02 (Based on CS 213 F’01) – 16 – class02.ppt Machine-Level Code Representation Encode Program as Sequence of Instructions Each simple operation –Arithmetic operation –Read or write memory –Conditional branch Instructions encoded as bytes –Alpha’s, Sun’s, Mac’s use 4 byte instructions »Reduced Instruction Set Computer (RISC) –PC’s use variable length instructions »Complex Instruction Set Computer (CISC) Different instruction types and encodings for different machines –Most code not binary compatible Programs are Byte Sequences Too!

17. CS 213 S’02 (Based on CS 213 F’01) – 17 – class02.ppt Representing Instructions int sum(int x, int y) { return x+y; } Different machines use totally different instructions and encodings 00 30 42 Alpha sum 01 80 FA 6B E0 08 81 C3 Sun sum 90 02 00 09 For this example, Alpha & Sun use two 4-byte instructions –Use differing numbers of instructions in other cases PC uses 7 instructions with lengths 1, 2, and 3 bytes –Same for NT and for Linux –NT / Linux not fully binary compatible E5 8B 55 89 PC sum 45 0C 03 45 08 89 EC 5D C3

18. CS 213 S’02 (Based on CS 213 F’01) – 18 – class02.ppt Boolean Algebra Developed by George Boole in 19th Century Algebraic representation of logic –Encode “True” as 1 and “False” as 0 And A&B = 1 when both A=1 and B=1 Not ~A = 1 when A=0 Or A|B = 1 when either A=1 or B=1 Exclusive-Or (Xor) A^B = 1 when either A=1 or B=1, but not both

19. CS 213 S’02 (Based on CS 213 F’01) – 19 – class02.ppt A ~A ~B B Connection when A&~B | ~A&B = A^B Application of Boolean Algebra Applied to Digital Systems by Claude Shannon 1937 MIT Master’s Thesis Reason about networks of relay switches –Encode closed switch as 1, open switch as 0

20. CS 213 S’02 (Based on CS 213 F’01) – 20 – class02.ppt Properties of & and | Operations Integer Arithmetic  Z, +, *, –, 0, 1  forms a “ring” Addition is “sum” operation Multiplication is “product” operation – is additive inverse 0 is identity for sum 1 is identity for product Boolean Algebra  {0,1}, |, &, ~, 0, 1  forms a “Boolean algebra” Or is “sum” operation And is “product” operation ~ is “complement” operation (not additive inverse) 0 is identity for sum 1 is identity for product

21. CS 213 S’02 (Based on CS 213 F’01) – 21 – class02.ppt Properties of Rings & Boolean Algebras Boolean AlgebraInteger Ring Commutativity A | B = B | AA + B = B + A A & B = B & AA * B = B * A Associativity (A | B) | C = A | (B | C)(A + B) + C = A + (B + C) (A & B) & C = A & (B & C)(A * B) * C = A * (B * C) Product distributes over sum A & (B | C) = (A & B) | (A & C)A * (B + C) = A * B + B * C Sum and product identities A | 0 = AA + 0 = A A & 1 = AA * 1 = A Zero is product annihilator A & 0 = 0A * 0 = 0 Cancellation of negation ~ (~ A) = A– (– A) = A

22. CS 213 S’02 (Based on CS 213 F’01) – 22 – class02.ppt Ring  Boolean Algebra Boolean AlgebraInteger Ring Boolean: Sum distributes over product A | (B & C) = (A | B) & (A | C)A + (B * C)  (A + B) * (B + C) Boolean: Idempotency A | A = AA + A  A –“A is true” or “A is true” = “A is true” A & A = AA * A  A Boolean: Absorption A | (A & B) = AA + (A * B)  A –“A is true” or “A is true and B is true” = “A is true” A & (A | B) = AA * (A + B)  A Boolean: Laws of Complements A | ~A = 1A + –A  1 –“A is true” or “A is false” Ring: Every element has additive inverse A | ~A  0A + –A = 0

23. CS 213 S’02 (Based on CS 213 F’01) – 23 – class02.ppt Properties of & and ^ Boolean Ring  {0,1}, ^, &, , 0, 1  Identical to integers mod 2  is identity operation:  (A) = A A ^ A = 0 PropertyBoolean Ring Commutative sumA ^ B = B ^ A Commutative productA & B = B & A Associative sum(A ^ B) ^ C = A ^ (B ^ C) Associative product(A & B) & C = A & (B & C) Prod. over sumA & (B ^ C) = (A & B) ^ (B & C) 0 is sum identityA ^ 0 = A 1 is prod. identityA & 1 = A 0 is product annihilatorA & 0 = 0 Additive inverseA ^ A = 0

24. CS 213 S’02 (Based on CS 213 F’01) – 24 – class02.ppt Relations Between Operations DeMorgan’s Laws Express & in terms of |, and vice-versa A & B = ~(~A | ~B) »A and B are true if and only if neither A nor B is false A | B = ~(~A & ~B) »A or B are true if and only if A and B are not both false Exclusive-Or using Inclusive Or A ^ B = (~A & B) | (A & ~B) »Exactly one of A and B is true A ^ B = (A | B) & ~(A & B) »Either A is true, or B is true, but not both

25. CS 213 S’02 (Based on CS 213 F’01) – 25 – class02.ppt General Boolean Algebras Operate on Bit Vectors Operations applied bitwise Representation of Sets Width w bit vector represents subsets of {0, …, w–1} a j = 1 if j  A –01101001 { 0, 3, 5, 6 } –01010101 { 0, 2, 4, 6 } & Intersection 01000001 { 0, 6 } | Union 01111101 { 0, 2, 3, 4, 5, 6 } ^Symmetric difference 00111100 { 2, 3, 4, 5 } ~Complement 10101010 { 1, 3, 5, 7 } 01101001 & 01010101 01000001 01101001 | 01010101 01111101 01101001 ^ 01010101 00111100 ~ 01010101 10101010

26. CS 213 S’02 (Based on CS 213 F’01) – 26 – class02.ppt Bit-Level Operations in C Operations &, |, ~, ^ Available in C Apply to any “integral” data type –long, int, short, char View arguments as bit vectors Arguments applied bit-wise Examples (Char data type) ~0x41 --> 0xBE ~01000001 2 -->10111110 2 ~0x00 --> 0xFF ~00000000 2 -->11111111 2 0x69 & 0x55 --> 0x41 01101001 2 & 01010101 2 --> 01000001 2 0x69 | 0x55 --> 0x7D 01101001 2 | 01010101 2 --> 01111101 2

27. CS 213 S’02 (Based on CS 213 F’01) – 27 – class02.ppt Contrast: Logic Operations in C Contrast to Logical Operators &&, ||, ! –View 0 as “False” –Anything nonzero as “True” –Always return 0 or 1 –Early termination Examples (char data type) !0x41 --> 0x00 !0x00 --> 0x01 !!0x41 --> 0x01 0x69 && 0x55 --> 0x01 0x69 || 0x55 --> 0x01 p && *p ( avoids null pointer access)

28. CS 213 S’02 (Based on CS 213 F’01) – 28 – class02.ppt Shift Operations Left Shift: x << y Shift bit-vector x left y positions –Throw away extra bits on left –Fill with 0’s on right Right Shift: x >> y Shift bit-vector x right y positions –Throw away extra bits on right Logical shift –Fill with 0’s on left Arithmetic shift –Replicate most significant bit on right –Useful with two’s complement integer representation 01100010 Argument x 00010000<< 3 00011000 Log. >> 2 00011000 Arith. >> 2 10100010 Argument x 00010000<< 3 00101000 Log. >> 2 11101000 Arith. >> 2

29. CS 213 S’02 (Based on CS 213 F’01) – 29 – class02.ppt Cool Stuff with Xor void funny(int *x, int *y) { *x = *x ^ *y; /* #1 */ *y = *x ^ *y; /* #2 */ *x = *x ^ *y; /* #3 */ } Bitwise Xor is form of addition With extra property that every value is its own additive inverse A ^ A = 0